Description

In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated both membership and typicality values when clustering unlabeled data. FPCM constrains the typicality values so that the sum over all data points of typicalities to a cluster is one. The row sum constraint produces unrealistic typicality values for large data sets. In this paper, we propose a new model called possibilistic-fuzzy c-means (PFCM) model. PFCM produces memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. PFCM is a hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) that often avoids various problems of PCM, FCM and FPCM. PFCM solves the noise sensitivity defect of FCM, overcomes the coincident clusters problem of PCM and eliminates the row sum constraints of FPCM. We derive the first-order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima of the PFCM objective functional. Several numerical examples are given that compare FCM and PCM to PFCM. Our examples show that PFCM compares favorably to both of the previous models. Since PFCM prototypes are less sensitive to outliers and can avoid coincident clusters, PFCM is a strong candidate for fuzzy rule-based system identification. The necessary conditions in for the PFCM model hold for any inner product norm, e.g., for the scaled Mahalanobis norm , so the formulation is quite general. The two main branches of generalization for c-means models are locally “adaptive” schemes such as those of Gustafson and Kessel or Dave and Bhaswan ; and extensions of the prototypes to shell-like surfaces, see for example Krishnapuram et al. . The basic architecture of the PFCM -AO algorithm will clearly remain the same, but the update equations for extensions in either direction will need to be modified by the appropriate necessary conditions. Many of these extensions will be straightforward, and we hope to write about some of them soon.